Note that eq:zq implies that \( (\mathcal {S}_{\alpha ,\tau }\mathcal {R}_{\theta _0}\cdot )(T-) \)  is a bounded linear operator from \( L^2(0,T;Z) \)  to \( Y\) . In addition, \( U_\text{ad} \)  is a convex, bounded and closed subset of \(L^2(0,T;Z) \) . Hence, applying [1]} to
problem eq:numeropti yields the following theorem, by
lem:Stau-dual.
